A well-posed integral equation formulation for three-dimensional rough surface scattering

Abstract

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage–Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space when the scattering surface does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, κ , for κ >0, if the coupling parameter η is chosen proportional to the wave number. In the case when is a plane, we show that the choice is nearly optimal in terms of minimizing the condition number.

Maxwell's equations in an unbounded structure

“…It follows from (5.24) that we obtain Z jr uj 2 The conclusion still holds if " satisfies the assumptions in [11]. Here, we give different assumptions on ", which are valid for a larger class of functions.…”

“…The scattering by unbounded structures has significant applications such as modeling acoustic and electromagnetic wave propagation over outdoor ground and sea surfaces or, at a very different scale, optical scattering from the surface of materials in near-field optics or nano-optics, detection of underwater mines, especially those buried in soft sediments. These problems are extensively studied and a considerable amount of information is available concerning their solutions [1][2][3][4][5][6][7][8][9].…”

Elastic scattering from rough surfaces in three dimensions

“…Significant progress has been made by Chandler-Wilde and his co-authors concerning the mathematical analysis and the numerical approximation of the acoustic scattering problems modeled by the Helmholtz equation. We refer to [10,11,14,15,41] for the integral equation method and to [8,12] the variational approach in both two and three dimensional settings. In the work of Duran, Muga and Nedelec [38], the radiation condition and well-posedness in the absence of acoustic surfaces waves were discussed under the non-absorbing boundary condition in a locally perturbed half plane.…”

Time-Harmonic Acoustic Scattering from a Nonlocally Perturbed Trapezoidal Surface

“…However, in the case of a globally perturbed rough surface, by which we mean a non-local perturbation of a planar surface such that the surface lies within a finite distance of the original plane, one in general cannot explicitly extract an outgoing wave from the scattered wave to meet the SRC. The Angular Spectrum Representation (ASR) radiation condition (see [18,9,10]) or the Upward Propagating Radiation Condition (UPRC) (see [41,42,12]) can be viewed as a rigorous formulation of a radiation condition to show the well-posedness of the problem in both two and three dimensions. The radiation condition relies also on the type of incoming waves in rough surface scattering problems.…”